What is Allometry?

• From greek: \(\small\acute\alpha\lambda\lambda\omicron\) (álo) = other, different + \(\small\mu\epsilon\tau\rho\acute\omega\) (metró) = to measure
• Differences in body proportions of an organism
• A departure from geometric similarity
• The relationship between size and shape

• Why?
• How? Types of allometry and hypothesis testing
• Visualizing allometric trends: summarizing multivariate patterns
• Custom tests: using permutations to approach biological questions

Why Allometry?

• Allometry occurs due to several reasons:
  • Physical (e.g. surface area/volume ratio)
  • Physiological
  • Biomechanical/functional
  • Selection (e.g. sexual selection and male ornaments)

Allometry: Basic Concepts

• What is allometry, and compared to what?
  • Isometry: no change in proportions
  • Allometry: change in proportions
• Easily envisioned via single trait relative to size:

\(\small{log(Y)}=log(\beta)+\alpha{log(x)}\)

Note: concepts of positive and negative allometry are only applicable to univariate traits! (for multivariate traits, allometry is a vector in multi-dimensional space where ‘positive’ and ‘negative’ as directions are not defined and have no meaning)

Also, it is mathematically impossible for all body parts to grow simultaneously faster (positive) or slower (negative) than total body size

Types of Allometry

• Allometric patterns may be observed at several levels of biological organization:
  • Ontogenetic allometry: throughout growth (e.g. understanding energy allocation)
  • Static allometry : within a single life stage (e.g. ornaments and sexual selection)
  • Evolutionary allometry: across species (e.g. adaptive shape changes)

Allometry in GM Data

• Shape: “the geometric properties of an object after standardizing the effects of location, scale and rotation”
• Procrustes shape variables are independent of CS when no allometry is present
• CS: mathematically independent of shape; biological allometric effects are still there!

Here is isometry…

Allometry in GM Data

• Shape: “the geometric properties of an object after standardizing the effects of location, scale and rotation”
• Procrustes shape variables are independent of CS when no allometry is present
• CS: mathematically independent of shape; biological allometric effects are still there!

Here is isometry… and here is allometry

• But how do we assess such patterns?

• We use linear models!

Multivariate Linear Regression

To characterize allometric patterns, we fit a model of shape~size as:

\[\small\mathbf{Y}=\mathbf{X}\mathbf{\beta } +\mathbf{E}\]

Multivariate Linear Regression

To characterize allometric patterns, we fit a model of shape~size as:

\[\small\mathbf{Y}=\mathbf{X}\mathbf{\beta } +\mathbf{E}\]

We evaluate the allometry model \(\small\mathbf{X}_{F}\) by comparing it to an intercept model \(\small\mathbf{X}_{R}\):

\(\tiny\mathbf{X}_R = \begin{bmatrix} 1\\ 1\\ 1\\ 1\\ 1\\ 1 \end{bmatrix}\) & \(\tiny\mathbf{X}_F = \begin{bmatrix} 1 & 0.3 \\ 1 & 0.5 \\ 1 & 0.2 \\ 1 & 1.2 \\ 1 & 0.7 \\ 1 & 1.1 \end{bmatrix}\)

Multivariate Linear Regression

To characterize allometric patterns, we fit a model of shape~size as:

\[\small\mathbf{Y}=\mathbf{X}\mathbf{\beta } +\mathbf{E}\]

We evaluate the allometry model \(\small\mathbf{X}_{F}\) by comparing it to an intercept model \(\small\mathbf{X}_{R}\):

\(\tiny\mathbf{X}_R = \begin{bmatrix} 1\\ 1\\ 1\\ 1\\ 1\\ 1 \end{bmatrix}\) & \(\tiny\mathbf{X}_F = \begin{bmatrix} 1 & 0.3 \\ 1 & 0.5 \\ 1 & 0.2 \\ 1 & 1.2 \\ 1 & 0.7 \\ 1 & 1.1 \end{bmatrix}\)

Assessing Significance: RRPP

1: Fit \(\small\mathbf{X}_{R}\) for each \(\small\mathbf{X}_{F}\); Estimate \(\small\hat{\mathbf{Y}}_{R}\) and \(\small\mathbf{E}_{R}\)

2: Permute, \(\small\mathbf{E}_{R}\): obtain pseudo-values as: \(\small\mathbf{\mathcal{Y}} = \mathbf{\hat{Y}}_{R} + \mathbf{E}_{R}\)

3: Fit \(\small\mathbf{X}_{F}\) using \(\small\mathbf{\mathcal{Y}}\): obtain coefficients and summary statistics

4: Calculate \(\small{F}\)-value in every random permutation (observed case counts as one permutation)

5: For \(\small{N}\) permutations, \(\small{P} = \frac{N(F_{random} \geq F_{obs})}{N}\)

6: Calculate effect size as a standard deviate of the observed value in a normalized distribution of random values (helps for comparing effects within and between models); i.e.: \[\small{z} = \frac{ \log\left( F\right) - \mu_{\log\left(F\right)} } { \sigma_{\log\left(F\right)} }\] where \(\small\mu_{\log\left(F\right)}\) and \(\sigma_{\log\left(F\right)}\) are the expected value and standard deviation from the sampling distribution, respectively.

Collyer et al. Heredity. (2015); Adams & Collyer. Evolution. (2016); Adams & Collyer. Evolution. (2018)

Regression: Example

Does body shape covary with size in Pecos pupfish?

This is a hypothesis of Allometry (shape~size covariation)

data(Pupfish)
Pupfish$logSize <- log(Pupfish$CS)
fit <- procD.lm(coords ~ logSize, SS.type = "I", 
                data = Pupfish, print.progress = FALSE, iter = 999) 
anova(fit)

## 
## Analysis of Variance, using Residual Randomization
## Permutation procedure: Randomization of null model residuals 
## Number of permutations: 1000 
## Estimation method: Ordinary Least Squares 
## Sums of Squares and Cross-products: Type I 
## Effect sizes (Z) based on F distributions
## 
##           Df       SS        MS     Rsq      F      Z Pr(>F)   
## logSize    1 0.014019 0.0140193 0.24886 17.229 4.4621  0.001 **
## Residuals 52 0.042314 0.0008137 0.75114                        
## Total     53 0.056333                                          
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Call: procD.lm(f1 = coords ~ logSize, iter = 999, SS.type = "I", data = Pupfish,  
##     print.progress = FALSE)

• But how does one visualize multivariate allometry??

Visualizing Multivariate Allometry

• Need to summarize size-related variation in shape
• Use some rotation/projection method

Several (complementary) solutions:

• Regression scores (Drake and Klingenberg 2008): projection of shape data on the allometric line \[\small\mathbf{s}=\mathbf{Y\beta^T(\beta^T\beta)^{-1/2}}\]

• PC1 of predicted values (Adams and Nistri 2010): PC rotation of Ŷ to summarize variation across the shape-size relationship \[\small\mathbf{P_1}=svd(\mathbf{\hat{Y}})=svd(\mathbf{X(\tilde{X}^T\tilde{X})^{-1}(\tilde{X}^T\tilde{Y})})\]

Visualization: Example

• Effects of human-mediated disturbance (urbanization) on Podarcis lizards
  • 5 urban and 5 rural populations
  • 359 individuals
  • 28 bilateral landmarks
• How does environmental disturbance influence development?
• Are size-shape relationships modified in urban populations?

Allometry Example: Podarcis

## 
## Analysis of Variance, using Residual Randomization
## Permutation procedure: Randomization of null model residuals 
## Number of permutations: 1000 
## Estimation method: Ordinary Least Squares 
## Sums of Squares and Cross-products: Type I 
## Effect sizes (Z) based on F distributions
## 
##            Df      SS       MS     Rsq       F      Z Pr(>F)   
## CS          1 0.05814 0.058139 0.09503 40.0581 7.7415  0.001 **
## type        1 0.03235 0.032350 0.05287 22.2892 6.6662  0.001 **
## CS:type     1 0.00610 0.006105 0.00998  4.2061 3.3590  0.001 **
## Residuals 355 0.51524 0.001451 0.84212                         
## Total     358 0.61183                                          
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Call: procD.lm(f1 = Y.gpa$coords ~ CS * type, data = gdf, print.progress = FALSE)

Allometry Example: Podarcis (Cont.)

Visualization

Post-Hoc Tests

• Use randomization methods to test custom hypotheses
• Do urban populations show higher deviations from their group’s trajectory? – a sign of developmental instability

Size-Shape Space

• Size effects often dominate shape variation, especially in ontogenetic series
• However, when other factors are present, PC1 does not necessarily relate directly to size-dependent variation (at least not to all of it)
• Usual statistical methods do not capture subtle ontogenetic modifications in trajectories

• Examine allometric variation in size-shape space

Size-Shape Space

• Derived from the PCA of shape variables + \(\small{ln}(Csize)\)
  • 1: Calculate shape variables and \(\small{ln}(Csize)\), and join them in a single data matrix
  • 2: Calculate PCA of this matrix (\(\small{SSPC1}\))
  • 3: Generate the common allometric component (CAC): the common size-shape regression coefficient, after mean-centering shapes by group \(\small\mathbf{Y}_{cent}\)
  • 4: Calculate the residual shape component: SVD of data orthogonal to CAC
  • 5: Visualize allometry as (CAC vs. SSPC1) & (CAC vs. RSC1)

\[\small\alpha=(\mathbf{Y}_{cent}^T\mathbf{CS})/\mathbf{(CS^TCS)}\] \[\small{CAC}=\mathbf{Y}_{cent}\alpha\] where \[\small\alpha=\alpha\sqrt{}(\alpha^T\alpha)\]

Size-Shape Space: Example

Note: CAC Scores vs. Regression Scores

• \(\small{CAC}\): regression under a common slope after group mean-centering
• Regression scores (\(\small{s}\)): shape data projected on the regression line with size
• When a single group is considered, \(\small{CAC}\) and \(\small{s}\) are identical!
• Both are projections of mean-centered data on size/shape regression

\[\small\alpha=(\mathbf{Y}_{cent}^T\mathbf{CS})/\mathbf{(CS^TCS)}\] \[\small{CAC}=\mathbf{Y}_{cent}\alpha\] \[\small\alpha=\alpha\sqrt{}(\alpha^T\alpha)\] \[\small\mathbf{s}=\mathbf{Y\beta^T(\beta^T\beta)^{-1/2}}\]

See Adams et al. 2013. Hystrix 24: 7-14.

Allometry: Other Tests

• Many biological hypotheses are related to the comparison of ontogenetic allometries
• Allometric trajectories are not only about their slope, or variation around it

Allometry: Other Tests

• Many biological hypotheses are related to the comparison of ontogenetic allometries
• Allometric trajectories are not only about their slope, or variation around it
• E.g. they have a starting and ending point
• Are ontogenetic trajectories similar? (parallelism)
• Do ontogenetic trajectories converge?
• Do ontogenetic trajectories diverge?

Allometry: Other Tests

• Many biological hypotheses are related to the comparison of ontogenetic allometries
• Allometric trajectories are not only about their slope, or variation around it
• E.g. they have a starting and ending point
• Are ontogenetic trajectories similar? (parallelism)
• Do ontogenetic trajectories converge?
• Do ontogenetic trajectories diverge?
• We need quantitative methods to test such hypotheses
• Take advantage of the toolkit of resampling methods

See also PTA for a general method for comparing all types of trajectories, including ontogenetic/allometric trajectories (i.e. Collyer and Adams 2007; Adams and Collyer 2009; Collyer and Adams 2013)

Allometric Convergence and Divergence

• Convergence: shape is more similar at the end of the trajectory, than at the beginning
• A simple approach: test angular difference between trajectories

Allometric Convergence and Divergence

• Convergence: shape is more similar at the end of the trajectory, than at the beginning
• A simple approach: test angular difference between trajectories
  • 1: Obtain allometric vectors - \(\small\mathbf{\hat{Y}}_{start}\) & \(\small\mathbf{\hat{Y}}_{end}\) - for groups
  • 2: Normalize vectors (divide by their length so they are unit length) and find the cross-product between pairs of vectors; i.e, \(r_{ij} = \mathbf{\beta}_i^T \mathbf{\beta}_j\)
  • 3: Obtain test measure: \(\theta = \cos^{-1}(r_{ij})\)
  • 4: RRPP with appropriate null model to test \(\theta\)
• Convergence when \(\theta\) is significant and vectors tend toward a similar shape
• Divergence when \(\theta\) is significant and vectors tend toward a different shape
• Parallelism when \(theta\) is not significant and/or effect size is small
• Can also test absolute difference in vector lengths to determine if one group has greater onotogenetic shape change than another

Adams and Nistri. (2010) BMC. Evol. Biol.

Evolutionary Convergence: Example

• Ontogenetic series for 8 species of Hydromantes
• These salamanders live in caves and they climb extensively on the walls as adults
• Foot shape is a trait with functional relevance
• How does foot shape change across ontogeny?

Evolutionary Convergence: Example (Cont.)

Evolutionary Convergence: Example (Cont.)

Comparing allometric trajectories: geomorph example

• Question: Do your groups have parallel allometries?
• Question: Do your groups have unique allometries?
• Question: Which groups have different allometries, and how do they differ?

Example: pupfish data

data(pupfish)
plotAllSpecimens(pupfish$coords)

Comparing allometric trajectories: geomorph example (cont.)

Example: pupfish data

pupfish$logSize <- log(pupfish$CS)
pupfish$Group <- interaction(pupfish$Pop, pupfish$Sex)
fit.common <- procD.lm(coords ~ logSize + Group, 
                     data = pupfish, print.progress = FALSE) 
fit.unique <- procD.lm(coords ~ logSize * Group, 
                     data = pupfish, print.progress = FALSE)  

Comparing allometric trajectories: geomorph example (cont.)

Example: pupfish data

Homogeneity of slopes test

anova(fit.unique, fit.null = fit.common, print.progress = FALSE)

## 
## Analysis of Variance, using Residual Randomization
## Permutation procedure: Randomization of null model residuals 
## Number of permutations: 1000 
## Estimation method: Ordinary Least Squares 
## Effect sizes (Z) based on F distributions
## 
##                                 ResDf Df      RSS         SS         MS
## coords ~ logSize * Group (Null)    46  1 0.022079                      
## coords ~ logSize + Group           49 -3 0.024084 -0.0020045 0.00066816
## Total                              53    0.056333                      
##                                       Rsq      F       Z     P Pr(>F)
## coords ~ logSize * Group (Null)  0.000000                            
## coords ~ logSize + Group        -0.035583 1.3594 -2.2247 0.988       
## Total

Results suggest that allometric slopes are parallel. We can see by plotting the prediction lines for both that a unique-allometries model is not practical.

Comparing allometric trajectories: geomorph example (cont.)

Example: pupfish data

Allometric plots

par(mfcol = c(1,2))
plotAllometry(fit.common, pupfish$CS, logsz = TRUE, method = "PredLine", pch = 21,
              bg = pupfish$Group)
plotAllometry(fit.unique, pupfish$CS, logsz = TRUE, method = "PredLine", pch = 21,
              bg = pupfish$Group)

Comparing allometric trajectories: geomorph example (cont.)

Example: pupfish data

Hypothesis tests: vector lengths

PW <- pairwise(fit.unique, groups = pupfish$Group,
               covariate = log(pupfish$CS), print.progress = FALSE)
summary(PW, test = "dist")

## 
## Pairwise comparisons
## 
## Groups: Marsh.F Sinkhole.F Marsh.M Sinkhole.M 
## 
## RRPP: 1000 permutations
## 
## Slopes (vectors of variate change per one unit of covariate change, by group):
## Vectors hidden (use show.vectors = TRUE to view)
## 
## Slope vector lengths
##    Marsh.F Sinkhole.F    Marsh.M Sinkhole.M 
## 0.09846857 0.08048951 0.08899635 0.13005765 
## 
## Pairwise absolute difference (d) between vector lengths, plus statistics
##                                 d  UCL (95%)          Z Pr > d
## Marsh.F:Sinkhole.F    0.017979061 0.08701897 -0.6684921  0.691
## Marsh.F:Marsh.M       0.009472222 0.08066795 -0.9266208  0.805
## Marsh.F:Sinkhole.M    0.031589079 0.08573929 -0.1404949  0.485
## Sinkhole.F:Marsh.M    0.008506839 0.04838718 -0.7095093  0.706
## Sinkhole.F:Sinkhole.M 0.049568140 0.04244181  2.4759957  0.023
## Marsh.M:Sinkhole.M    0.041061301 0.04776048  1.3887438  0.093

Comparing allometric trajectories: geomorph example (cont.)

Example: pupfish data

Hypothesis tests: angles between vectors

summary(PW, test = "VC", angle.type = "deg") 

## 
## Pairwise comparisons
## 
## Groups: Marsh.F Sinkhole.F Marsh.M Sinkhole.M 
## 
## RRPP: 1000 permutations
## 
## Slopes (vectors of variate change per one unit of covariate change, by group):
## Vectors hidden (use show.vectors = TRUE to view)
## 
## Pairwise statistics based on slopes vector correlations (r) and angles, acos(r)
## The null hypothesis is that r = 1 (parallel vectors).
## This null hypothesis is better treated as the angle between vectors = 0
##                               r    angle UCL (95%)          Z Pr > angle
## Marsh.F:Sinkhole.F    0.6318267 50.81498  82.18877 -0.2229753      0.538
## Marsh.F:Marsh.M       0.6139591 52.12367  85.99703 -0.2996071      0.582
## Marsh.F:Sinkhole.M    0.4461018 63.50614  82.21052  0.5581365      0.260
## Sinkhole.F:Marsh.M    0.7175129 44.15048  65.21340  0.1341814      0.395
## Sinkhole.F:Sinkhole.M 0.4627734 62.43378  59.87174  2.1810996      0.036
## Marsh.M:Sinkhole.M    0.5629975 55.73665  66.72786  1.0332053      0.146

Comparing allometric trajectories: geomorph example (cont.)

Example: pupfish data

Conclusion: allometries do not significantly diverge. But concluding they are parallel allows one to compare means, accounting for common allometric variation.

PW <- pairwise(fit.common, groups = pupfish$Group,
               print.progress = FALSE)

summary(PW, test = "dist") 

## 
## Pairwise comparisons
## 
## Groups: Marsh.F Sinkhole.F Marsh.M Sinkhole.M 
## 
## RRPP: 1000 permutations
## 
## LS means:
## Vectors hidden (use show.vectors = TRUE to view)
## 
## Pairwise distances between means, plus statistics
##                                d  UCL (95%)        Z Pr > d
## Marsh.F:Sinkhole.F    0.03639758 0.01731029 7.510885  0.001
## Marsh.F:Marsh.M       0.03579214 0.01849114 6.875381  0.001
## Marsh.F:Sinkhole.M    0.03809715 0.01593951 9.197696  0.001
## Sinkhole.F:Marsh.M    0.03681731 0.02334774 4.868793  0.001
## Sinkhole.F:Sinkhole.M 0.01939739 0.01845710 2.166170  0.033
## Marsh.M:Sinkhole.M    0.02694896 0.01879110 4.077037  0.001

Empirical Example: Cranial Ontogeny in Crocodilians

• Phylogenetic relationships have long been under debate
• What information can we extract from cranial ontogeny?

Empirical Example: Cranial Ontogeny in Crocodilians (Cont.)

• Some groups follow different ontogenetic paths
• There is ontogenetic convergence, where adults are more similar than juveniles

A Comment on Heterochrony

• Evolutionary changes in the rate or timing of developmental sequences
  • i.e. changes in ontogenetic trajectories
• Can be evaluated via allometric/ontogenetic investigations

Allometry: Summary

• Shape usually changes with size – allometry is a common pattern in biological data
• We need to account for it when testing hypotheses about shape
• Many biological hypotheses related directly to allometric variation
• Visualization:
  • RegScore (\(\small{s}\)) vs. Csize (Drake and Klingenberg 2008)
  • PredLine \(\small{P_1}\) vs. Csize (Adams and Nistri 2010)
  • CAC, RSC, Csize (Mitteroecker et al. 2004)
• Comparison of ontogenies: - MANCOVA
  • Convergence & divergence (Adams and Nistri 2010)
  • Orientation (Piras et al. 2010)
• Heterochrony and related topics also of interest